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Properties

Label 1-185-185.103-r0-0-0
Degree $1$
Conductor $185$
Sign $0.999 + 0.00214i$
Analytic cond. $0.859136$
Root an. cond. $0.859136$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s − i·6-s + (−0.866 + 0.5i)7-s + 8-s + (0.5 − 0.866i)9-s − 11-s + (0.866 + 0.5i)12-s + (−0.5 − 0.866i)13-s − i·14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s − i·6-s + (−0.866 + 0.5i)7-s + 8-s + (0.5 − 0.866i)9-s − 11-s + (0.866 + 0.5i)12-s + (−0.5 − 0.866i)13-s − i·14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.999 + 0.00214i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.999 + 0.00214i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4317918947 + 0.0004635982253i\)
\(L(\frac12)\) \(\approx\) \(0.4317918947 + 0.0004635982253i\)
\(L(1)\) \(\approx\) \(0.4943843372 + 0.1653661128i\)
\(L(1)\) \(\approx\) \(0.4943843372 + 0.1653661128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + T \)
29 \( 1 - iT \)
31 \( 1 + iT \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + T \)
47 \( 1 - iT \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.38849829092125154881935478247, −26.398272516211828330326391206112, −25.609347099780548790273441786566, −24.14660297782334453420725164843, −23.20721061125224130742798258236, −22.40717743103508625568124695450, −21.48074028850703086719105946472, −20.42504159467722245437127619619, −19.119697897982126276918192746222, −18.81724549998788388380551859861, −17.57139767610611162067026678424, −16.76079603735358375192394895889, −16.00478002904124878448178335863, −14.00819188481984965784918794240, −12.914888017590057795937387868660, −12.398856614927896060285115414601, −11.15887797547501425565815750284, −10.36129712521815407278790956778, −9.410441514762575400548979774793, −7.817235069310245896707839034265, −7.00310301522030245553680308513, −5.50644465902825235378897162531, −4.13206629436340675711999609454, −2.68353192431680474180874847818, −1.18412117782111322389248027937, 0.52944610858482258554387174180, 3.031750606369785331308382303065, 4.951360741425936506975483150857, 5.52100109700646038061016984367, 6.69798360826650068712615775247, 7.75613500613517705850802547603, 9.29658554250514349652545900633, 9.93740907413816604838915112775, 10.96560125453511220501960484915, 12.37326267312078059400751755489, 13.42766547871156976065866871729, 14.971343824116047413571944114925, 15.76247511790596310972868159422, 16.30912235328976451499211594158, 17.47846142516469969847234424648, 18.218910444621665105345929840594, 19.15495479684758013654574135844, 20.47041056190058757745740501319, 21.7704140424344993448299413526, 22.79323057815672262722125075407, 23.19737522954109688292528744567, 24.489699899452745885065200955366, 25.30677016299407561396551296292, 26.44256151936129004756045925606, 27.05563030814564381333552308066

Graph of the $Z$-function along the critical line